Question 7 - A-Level Maths

Express $5 \cos \theta - 12 \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving the value of $\alpha$ correct to 2 decimal places.
Hence solve the equation $5 \cos \theta - 12 \sin \theta = 8$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.
Find the greatest possible value of $$7 + 5 \cos \frac { 1 } { 2 } \phi - 12 \sin \frac { 1 } { 2 } \phi$$ as $\phi$ varies, and determine the smallest positive value of $\phi$ for which this greatest value occurs.
[0pt] [4]